GAMS — The Solver Manuals - Available Software

SNOPT 537
LU density tolerance r 1
LU singularity tolerance r 2
The density tolerance r 1 is used during LUSOLs basis factorization B = LU. Columns of L and rows of U
are formed one at a time, and the remaining rows and columns of the basis are altered appropriately. At
any stage, if the density of the remaining matrix exceeds r 1 , the Markowitz strategy for choosing pivots is
terminated and the remaining matrix is factored by a dense LU procedure. Raising the density tolerance
towards 1.0 may give slightly sparser LU factors, with a slight increase in factorization time.
The singularity tolerance r 2 helps guard against ill-conditioned basis matrices. After B is refactorized, the
diagonal elements of U are tested as follows: if |U jj | ≤ r 2 or |U jj | < r 2 max i |U ij |, the jth column of the
basis is replaced by the corresponding slack variable. (This is most likely to occur after a restart)
Default: LU density tolerance 0.6 and LU singularity tolerance 3.2e-11 for most machines. This
value corresponds to ɛ 2/3 , where ɛ is the relative machine precision.
Major feasibility tolerance ɛ r
This specifies how accurately the nonlinear constraints should be satisfied. The default value of 1.0e-6 is
appropriate when the linear and nonlinear constraints contain data to about that accuracy.
Let rowerr be the maximum nonlinear constraint violation, normalized by the size of the solution. It is
required to satisfy
rowerr = max viol i /‖x‖ ≤ ɛ r , (32.8)
i
where viol i is the violation of the ith nonlinear constraint (i = 1 : nnCon, nnCon being the number of
nonlinear constraints).
In the GAMS/SNOPT iteration log, rowerr appears as the quantity labeled “Feasibl”. If some of the
problem functions are known to be of low accuracy, a larger Major feasibility tolerance may be appropriate.
Default: Major feasibility tolerance 1.0e-6.
Major iterations limit k
This is the maximum number of major iterations allowed.
number of linearizations of the constraints.
Default: Major iterations limit max{1000, m}.
It is intended to guard against an excessive
Major optimality tolerance ɛ d
This specifies the final accuracy of the dual variables. On successful termination, SNOPT will have computed
a solution (x, s, π) such that
maxComp = max Comp j /‖π‖ ≤ ɛ d , (32.9)
j
where Comp j is an estimate of the complementarity slackness for variable j (j = 1 : n + m). The values
Comp j are computed from the final QP solution using the reduced gradients d j = g j − π T a j (where g j is the
jth component of the objective gradient, a j is the associated column of the constraint matrix ( A − I ),
and π is the set of QP dual variables):
{
dj min{x j − l j , 1} if d j ≥ 0;
Comp j =
−d j min{u j − x j , 1} if d j < 0.
In the GAMS/SNOPT iteration log, maxComp appears as the quantity labeled “Optimal”.
Default: Major optimality tolerance 1.0e-6.
Major print level p
This controls the amount of output to the GAMS listing file each major iteration. This output is only visible
if the sysout option is turned on (see §4.1). Major print level 1 gives normal output for linear and
nonlinear problems, and Major print level 11 gives additional details of the Jacobian factorization that
commences each major iteration.
In general, the value being specified may be thought of as a binary number of the form
Major print level JFDXbs
where each letter stands for a digit that is either 0 or 1 as follows: